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Fractals, Vol. 15, No. 1 (2007) 97–100c© World Scientific Publishing Company

MINORITY AND MAJORITY GAMESIN FINANCIAL MARKETS

KYUNGSIK KIM,∗,¶ SEONG-MIN YOON,† SOO YONG KIM‡ and HIDEKI TAKAYASU§∗Department of Physics and

†Division of EconomicsPukyong National University

599-1 Daeyeon-dong, Pusan 608-737, Korea‡Department of Physics, Korea Advanced Institute of Science and Technology

Daejeon 305-701, Korea§Sony Computer Science Laboratories

3-14-13 Higashi-gotanda, Shinagawa-Ku, 141-0022 Tokyo, Japan¶kskim@pknu.ac.kr

AbstractWe study the game theory from tick data of the won-dollar and yen-dollar exchange rates infinancial markets. The standard deviation, the global efficiency, and the autocorrelation forarbitrary strategies are shown to give rise to properties of new dynamics, and these statisticalquantities are very similar with the case of the majority game. Our results presented will becompared with numerical findings for other game models.

Keywords : Minority and Majority Games; Standard Deviation; Won-Dollar and Yen-DollarExchange Rates.

1. INTRODUCTION

For the last two decades, the dynamics of gametheories has received considerable attention asan interdisciplinary topic among physicists and

economists.1,2 Indeed, there has been intense focuson calculating and simulating game theories invarious ways such as the evolutionary minor-ity game,3–5 the adaptive minority game,6 the

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98 K. Kim et al.

multi-choice minority game,7 the $-game model,8

and the grand canonical minority games amongmany outstanding topics. Inspired by Challet andZhang,9 it is found that the stock prices are charac-terized by anomalous fluctuations and exhibited thefat tailed distribution and long range correlation.10

Using the minority and majority games and the$-game, the dynamical behavior of statistical quan-tities between different types of traders is studied.11

Moreover, it is showed from previous works12–14

that the grand canonical minority game reveals theimportant stylized facts of financial market phe-nomenology such as return changes and volatilityclusterings.

The Arthur’s bar model, the seller and buyermodel, and the passenger problem in the metroand bus have recently been treated mainly frommany researchers. The various models have dis-cussed several ways of rewarding one agent’s strate-gies and comparing with the resulting behaviorsof the configurations in the minority game the-ory. de Almeida and Menche15 investigated twooptions rewarded in standard minority games byusing adaptive genetic algorithms, and their resultwas found to come close to that of standard minor-ity games. Kim et al.16 analyzed the minority gamefor patients that is inspired by de Almeida andMenche’s problem, and in particular they appliedthe minority game theory to the numbers of bothgeneral patients and reserved patients. Mosettiet al.17 studied mainly the timescale heterogeneitiesfor the strategy change frequency, strategy-strategycorrelation, reaction rates and score memory in theminority game. Bianconi et al.18 also found thatthe likelihood of agents trading in an asset dependrelatively on the available amount of informationin the canonical and the grand-canonical minoritygames. Very recently Yoon et al.19 presented theminority game theory for the transaction numbersof two types of Korean treasury bond futures in theKorean futures exchange market.

The game theory has extensively applied to novelinvestigations for universal properties based on sta-tistical concepts and methods in financial models.It is of fundamental importance to estimate numeri-cally and analytically the statistical quantities usingthe game theory. Very recently, such an attemptin this direction has been made from the transac-tion numbers of two types of Korean treasury bondfutures in the Korean futures exchange market.19

In this paper, we focus on the minority game the-ory for the transaction numbers of the won-dollar

and yen-dollar exchange rates in financial exchangemarkets and discuss market payoffs and statisticalquantities in the game theory. We also estimatenumerically standard deviation, global efficiency,and autocorrelation for particular strategies usingthe minority game payoff. In Sec. 2, we discussthe dynamics of minority and majority games, andnumerical results and concluding remarks are givenin Sec. 3.

2. MINORITY AND MAJORITYGAMES

In this section, let us introduce the dynamical mech-anism of both the minority and majority games.We assume that the agents N can decide inde-pendently whether to buy or to sell the stock atround m. When the information and the strategytake, respectively, the value µ (µ ∈ [1, 2, . . . , P ])and s among the 22m

strategies at time t, the actionof the ith agent is presented in terms of ai,s

µ(t).One agent can submit an order ai,s

µ(t) = 1 (buy)or ai,s

µ(t) = −1 (sell), and the aggregate value,i.e. the sum of all action of agents, is given byA(t) =

∑Ni=0 ai,s

µ(t). The payoffs Ui,smg(t) and

Ui,smj(t) for minority and majority game models

are, respectively, represented in terms of

Ui,smg(t + 1) = Ui,s

mg(t) − ai,sµA(t) (1)

and

Ui,smj(t + 1) = Ui,s

mj(t) + ai,sµA(t). (2)

In order to evaluate numerically the payoff, weassume that N agents choose one among two pos-sible options, i.e. 1 or −1 (buy and sell) at eachround m. The agents obtain the score +1 (−1) ifthe return belongs to be smaller (larger) than zero.In our case, the agent’s action behaves indepen-dently without any communication or interaction.All available information is other agent’s actions,i.e. the memory of the past m rounds. From theaggregate value A(t), the standard deviation of A(t)is defined by

σ =

[1N

N∑i=1

〈(A2(t)〉]1/2

, (3)

where the bar denotes the average taken over con-figurations. Hence the volatility σ2 and the globalefficiency σ2/N can be found from the aggregatevalue. The functional form for the standard devia-tion as a function of the round m in the majority

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Minority and Majority Games in Financial Markets 99

game generally decreases as the round m increases,while the standard deviation generally has an opti-mal value in the minority game model.

The autocorrelation of A(t) is given by 〈A(t)A(t+1)〉, which decreases as t goes to a large valuein the majority game model.9 In financial markets,each agent plays as a majority game player and itspayoff is given by Eq. (2). In the $-game, the numer-ical behavior of the statistical quantities such asautocorrelation and volatility follows the minorityor majority game.

3. NUMERICAL RESULTS ANDCONCLUDING REMARKS

Our model can extend to several values of strat-egy, but we limit ourselves to only two strategies,s = 2 and s = 4. We introduce the price time seriesfor two financial assets as follows: we used the tickdata for the won-dollar exchange rate (yen-dollarexchange rate), which were taken from April 1981 toDecember 2002 (from January 1971 to June 2003).In our case, the total tick data are about N = 5953(8522) for the won-dollar exchange rate (yen-dollarexchange rate), respectively. In our case, the timestep between ticks for two assets is evoluted forone day.

First let us concentrate on calculations ofthe payoff and the standard deviation. At a time t,the agents are randomly selected by the history ofthe price return, R(t) = ln y(t + 1)/y(t), up to theround m before for the won-dollar and yen-dollarexchange rates. Next we introduce the standardstrategies that one decides to buy or sell among 22m

possible states of history in total. In our cases ofs = 2 and 4, we can find the payoff and the standarddeviation for the round m from an order ai,s

µ(t) = 1(buy) or ai,s

µ(t) = −1 (sell) when the price returnR(t) > 0 or R(t) < 0.

For the won-dollar and yen-dollar exchange ratesin financial exchange markets, we evaluate numeri-cally the standard deviation σ versus m (Fig. 1) forthe strategy s = 2, after the volatility from Eq. (3)is found numerically. Our result is expected tobehave in a way which is similar to the minor-ity game, but the dynamical behaviors for stan-dard deviation and the global efficiency are foundto be similar to the patterns of the majority game.The global efficiency is found to take the decreas-ing value near zero as the round m goes to a largevalue, as shown in Fig. 2. As shown in Fig. 3, thevalue of autocorrelations also decreases as m goes

2 120.000

1053 4 6 7 8 9 11

0.004

0.001

0.002

0.003

Yen-Dollar

Won-Dollar

σ

m

Fig. 1 Standard deviations versus a function of the roundm for a strategy s = 2 for the won-dollar and yen-dollarexchange rates.

-4 -2 0-1-3

Yen-Dollar

Won-Dollar

-8

-12

-13

-11

-10

-9

Fig. 2 Plot of global efficiencies σ2/N for the strategy s = 2for the won-dollar and yen-dollar exchange rates, where mmeans the round.

to a large value, and this dynamical behaviors aresimilar to those of the majority game. For s = 4,the standard deviation and the global efficiency arealso found to take similar forms of Figs. 1 to 3.

In conclusion, we found that statistical quan-tities such as the standard deviation, the globalefficiency, and the autocorrelation are very simi-lar with the case of the majority game. It is alsoshowed that a large number of agents are play-ing as majority game players, and that the pricedynamics follow a trend by majority game play-ers. Our findings are really similar to those of theKorean treasury bond futures,19 significantly dif-ferent with the result of minority game models.8

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100 K. Kim et al.

0

1 × 10-6

-4 -2 0-1-3-1 × 10-6

Yen-Dollar

Won-Dollar

< A

tA

t+1

>

Fig. 3 Autocorrelation of A(t) plot for the won-dollar andyen-dollar exchange rates, and this follows the functionalform of the majority game.

In order to assess the dynamical behavior for theminority game, majority game, and $-game, we canobtain statistical quantities such as standard devi-ation, volatility, autocorrelation, self-overlap, andentropy, etc. It is really expected that these statis-tical quantities lead us to more general results. Infuture, we expect that finance theories can enhancethe interpretation of results by the minority gametheory-based data analysis. Further investigationscan be conducted in other foreign exchange ratessuch as euro-dollar and dollar-mark exchange rates.

ACKNOWLEDGMENTS

This work was supported by Grand No. R01-2006-000-10470-0 from the Basic Program of the KoreaScience and Engineering Foundation.

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